Individual decision making
under risk
St. Petersburg paradox
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Consider a gamble x in which one of n outcomes will
occur
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And these n outcomes are worth respectively x1 , x2 , ..., xn
dollars
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The respective probabilities of these outcomes
are
P
p1 , p2 , ..., pn , where each pi ∈ [0, 1] and ni=1 pi = 1
How much is it worth to participate in this gamble?
P
The monetary expected value is Ex = ni=1 xi pi
One argument says it is the fair price of a gamble.
D. Bernoulli - St. Petersburg paradox:
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A fair coin is toosed until a head appears.
You receive 2n dollars if the first head occurs on trial n
How much are you willing to pay for such gamble?
What is the expected value of this gamble?
Bernoulli’s resolution
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Expected value of this gamble is infinite, and people
rarely want to give more than 10$ for it.
Bernoulli suggested the following modification to
expected value principle.
Instead of monetary value he proposed to use intrinsic
worths of these monetary values.
Intrinsic worth of money increases with money, but at a
diminshing rate.
A function with this property is the logarithm
If the "utility" of m dollars is log m, then the fair price
would not be monetary expected value but the monetary
equivalent of the utility expected value:
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Utility expected value E log x = 21 log 2 +
Monetary equivalent CE (x) = exp(E log x)
1
4
log 4 + ...
Criticism of Bernoulli’s proposal
There are certain obvious criticism of Bernoulli’s tack:
I The utility association to money is completely ad hoc.
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There are many strictly increasing concave functions.
Why should a decision be based upon the expected
value of these utilities?
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Rationale for expected value is usually based on the
effect of repreating the gamble many times.
Why should it extend to the situation in which a gamble
is played only once?
Von Neumann and Morgenstern
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What we want is a construction of a utility function for
each individual which in some sense represents her
choices among gambles.
And which has as a consequence the fact that the
expected value of utility represents the utility of the
corresponding gamble
Von Neumann and Morgenstern have shown the
following:
Theorem
If a person is able to express preferences between every
possible pair of gambles, where the gambles are taken
over some basic set of laternatives, then one can introduce
utility associations to the basic alternatives in such a
manner that, if the person is guided solely by the utility
expected value, he is acting in accord with his true
tastes - provided only that there is an element of
consistency in his tastes.
vNM utility function
Two points about this theorem and what is called von
Neumann Morgenstern utility function:
a) Reflects preferences about the alternatives in a certain
given situation.
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e.g. the resulting function will incorporate attitude
towards the whole gambling situation.
b) Justifies the central role of expected value without any
further argument
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Specifically, without any discussion of long run effects.
The essence of the idea
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Suppose that an individual reveals: A B, B C and
A C.
Ordinal utility - any three numbers in decreasing in
magnitude
But here we admit gambles
Suppose we ask his preferences between:
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obtaining B for certain
a gamble (A, p; C , 1 − p)
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If p is sufficiently close to 1, then a gamble will be
preferred
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If p is near 0, then the certain option will be preferred
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There has to be p, such that he is indifferent between
the two
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We suppose that there is exactly one such p ∈ (0, 1), let it
be 32
The essence of the idea
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If we arbitrarily associate 1 to alternative A and 0 to C
Then what number should we associate to B?
Naturally, 23
If this choice is made, then B is fair equivalent for the
gamble (A, 23 ; C , 31 )
in the sense that the utility of B equals the utility
expected value of the gamble
1
2
2
1× +0× =
3
3
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There are triples of numbers other than (1, 23 , 0) which
also reflect the same preferences as
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(a + b, a + b, b), wherea > 0
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Nevertheless, we are not allowed to say that going from
B to C is twice as desirable than going from A to B
Consistency demands
1) Any two alternatives should be comparable
2) Preference relations for lotteries are transitive
3) If a lottery has as one of its alternatives another lottery,
then the first lottery is decomposable into the more
basic alternatives through the use of the probability
calculus
4) If two lotteries are indifferent to the subject, then they
are interchangable as alternatives in any compound
lottery
5) If two lotteries involve the same two alternatives, then
the one in which the more preferred altenative has a
higher probability of occuring is itself preferred
6) If A is preferred to B and B to C , then there exists a
lottery involving A and C (with apropriate probabilities)
which is indifferent to B
Formal postulates/assumptions/axioms
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All loteries are build up from a finite set of basic
alternatives A1 , A2 , ..., An
We assume w.l.o.g. that A1 % ... % An and A1 An .
P
L = (A1 , p1 ; ..., An , pn ), where 0 ≥ pi ≥ 1 and ni=1 pi = 1 is
a typical lottery
Objective probabilities and yet the lottery played once
only (we do not view lottery from a frequency point of
view)
Assumption (1)
The weak preference relation % defined over the space of
basic alternatives is complete and transitive
Reduction of compound lotteries
Assumption (2 reduction of compound lotteries)
If L(i) = (A1 , p1 ; ..., An , pn ), for i = 1, 2, ..., s, then
(1)
(n)
(L(1) , q1 ; ...; L(s) , qs ) ∼ (A1 , p1 ; ...; An , pn )
where pi = q1 pi + ... + qs pi .
(1)
(s)
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Experiments such as p(1) and q are statistically
independent
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Or numbers such as pj actually denotes the conditional
probability of prize j in experiment p(i) given that lottery
i arose from experiment q
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(i)
The assumption abstracts away all "joy in gambling",
"atmosphere of the game", "pleasure in suspense"
IIR
Assumption (3 continuity)
For each Ai , there exists a number ui such that
Ai ∼ [A1 , ui ; An , 1 − ui ] = Ãi .
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Critics emphasize extreme events such as death
But it is not really a problem
Assumption (4 Substitutibility or IIA)
For any lottery L:
(A1 , p1 ; ...; Ai , pi ; ...; An , pn ) ∼ (A1 , p1 ; ...; Ãi , pi ; ...; An , pn )
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This assumption, taken with the third, says that not only
Ai is indifferent to Ãi as taken alone, but also when
substituted in any lottery ticket.
Two final assumptions
Assumption (5 transitivity)
Preference among lottery tickets is a transitive relation
Assumption (6 monotonicity)
A lottery (A1 , p; An , 1 − p) is weakly preferred to
(A1 , p 0 ; An , 1 − p 0 ) if and only if p ≥ p 0
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But a mountain climber prefers "life" to "death" and also
some lottery of life and death to life itself
However the real alternative is the whole gestalt of
climbing
Sometimes we need richer set of alternatives for this
assumption to be valid
Expected Utility Theorem
Theorem
If prefererence relation % satisfies assumptions 1-6, then
there are numbers ui associated with the basic prizes Ai
such that for two lotteries L and L0 the magnitudes of the
expected values
p1 u1 + ... + pn un and p10 u1 + ... + pn0 un
reflect the preference between the lotteries.
Example
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Let L = (A1 , 0.25; A2 , 0.25; A3 , 0.25; A4 , 0.25) and
L0 = (A1 , 0.15; A2 , 0.50; A3 , 0.15; A4 , 0.20)
Which one should I choose?
Suppose we determine that A2 ∼ (A1 , 0.6; A4 , 0.4) and
A3 ∼ (A1 , 0.2; An , 0.8)
Proof
1 Let L = (A1 , p1 ; ...; An , pn ) and L0 = (A1 , p10 ; ...; An , pn0 ) be any
two lotteries.
2 By Assumption (2) we can always write lotteries in this
basic way
3 Replace each Ai in lottery L by Ãi . Assumption (3) says
these indifferent elements exist and assumption (4) says
they are substitutable.
4 By using transitvity of indifference serially, we get
(A1 , p1 ; ...; An , pn ) ∼ (Ã1 , p1 ; ...; Ãn , pn )
5 If we sequentially apply assumption (2), we get:
(A1 , p1 ; ...; An , pn ) ∼ (A1 , p; An , 1 − p)
where p = p1 u1 + ... + pn un
6 Do steps 3-5 for lottery L0 to arrive at
p 0 = p10 u1 + ... + p20 u2 .
7 Finally, by assumption (6) L % L0 ⇐Ñ p ≥ p 0
Ordinal vs cardinal utility
Lottery
u
u0
A1
1.0
1.6
A2
0.6
0.8
Lottery
v
v0
A1
1.0
16
A2
0.6
15.9
A3
0.2
3.0
A4
0.0
2.0
A2
0.6
4.0
A3
0.2
1.0
A4
0.0
2.2
Lottery
s
s0
A1
1.0
3.0
A3
0.2
0.0
A4
0.0
−0.4
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Cardinal von Neumann Morgenstern utility function is
unique up to affine transformation
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Ordinal utility function is unique up to any strictly
monotonic transformation
Some common falacies
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The decision maker behaves as if he were a maximizer
of expected values of utility
Some common falacies
Falacy (1)
Lottery (A1 , p1 ; ...; An , pn ) is preferred to lottery
(A1 , p10 ; ...; An , pn0 ) because the utility of the former,
p1 u1 + ... + pn un , is greater than the utility of the latter,
p10 u1 + ... + pn0 un .
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We only devise a convenient way to represent
preferences
Preferences among lotteries logically precede the
introduction of utility function
Some common falacies
Falacy (2)
Suppose that A B C D and that the utilities of these
alternatives satisfy u(A) + u(D) = u(B) + u(C ), then
(B, 12 ; C , 12 ) should be preferred to (A, 21 ; D, 12 ), because
although they have the same expected utility, but the
former has smaller variance.
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Utility function reflects completely a person’s
preferences among risky alternatives
Falacy (3)
Suppose that A B C D and that the utilities of these
alternatives satisfy u(A) − u(B) > u(C ) − u(D), then the
change from B to A is more preferred than the change
from D to C
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Utility function is constructed from preferences between
pairs of alternatives, not between pairs of pairs of
alternatives
Some common falacies
Falacy (4)
Interpersonal comparisons of utility
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Utility function is unique up to affine transformation
We can choose a zero and a unit as we please
Zero is not a problem but arbitrary unit is
Suppose two people are isolated from each other and
each is given a measuring stick with some (possibly
different) unit of measurement
One subject is given full scale plans of a building
He is permitted to send only angles and lengths (in his
units of measurement) to the other guy
Conclusion: There is no natural zero in utility measurement.
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In the ordinal utility representation for choice under
certainty the crucial axiom "giving" order is
TRANSITIVITY.
Completeness is more a "technical" assumption needed
in order to be able to make comparisons
Here in the cardinal representation for choice under
risk the crucial axiom "giving" cardinality is
INDEPENDENCE
Continuity is merely a technical assumption needed for
existence of utility associations
Monotonicity (which gives uniqueness of utility
associations) and reduction of compound lotteries are
not needed since they follow from other axioms in the
general case
On the next slide the more general formulation is
presented
Axioms of von Neumann and Morgenstern
Axiom (Weak order)
% is complete and transitive
Axiom (Continuity)
For every P, Q, R ∈ L,
P Q R Ñ ∃α, β ∈ (0, 1) : αP +(1−α)R Q βP +(1−β)R
Axiom (Independence)
For every P, Q, R ∈ L, and every α ∈ (0, 1),
P % Q ⇐Ñ αP + (1 − α)R % αQ + (1 − α)R
Expected Utility Theorem
Theorem (vNM)
%⊂ L × L satisfies Axioms 1-3 if and only if there exists
u : X Ï R such that, for every P, Q ∈ L
X
X
P % Q ⇐Ñ
P(x)u(x) ≥
Q(x)u(x)
x∈X
x∈X
Moreover, in this case u is unique up to a positive linear
transformation.